( * 20 c n 0 ) + 2 , ( * 20 c n 1 ) + 2^2 ( * 20 c n 2 ) + ..... … - Vidyadip

MCQ

$\left( {\begin{array}{*{20}{c}}n\\0\end{array}} \right) + 2\,\left( {\begin{array}{*{20}{c}}n\\1\end{array}} \right) + {2^2}\left( {\begin{array}{*{20}{c}}n\\2\end{array}} \right) + ..... + {2^n}\left( {\begin{array}{*{20}{c}}n\\n\end{array}} \right)$ is equal to

A
${2^n}$
B
$0$
✓
${3^n}$
D
None of these

✓

Answer

Correct option: C.

${3^n}$

c
(c) ${(1 + x)^n} = {}^n{C_0} + x.{}^n{C_1} + {x^2}.{}^n{C_2} + .... + {x^n}.{}^n{C_n}$

Put $x = 2$

==> ${3^n} = {}^n{C_0} + 2.{}^n{C_1} + {2^2}.{}^n{C_2} + {2^3}.{}^n{C_3} + .... + {2^n}{.^n}{C_n}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $f(x) = \int_0^x {t(\sin \,\,x\, - \sin \,\,t)\,dt} $ then ?

Let $f: R \rightarrow R$ be a function defined by $f(x)=\max \left\{x, x^{2}\right\} .$ Let $S$ denote the set of all points in $R ,$ where $f$ is not differentiable. Then

Let $\vec u\;$be a vector coplanar with the vector $\vec a = 2\hat i + 3\hat j - \hat k$ and $\vec b = \hat j + \hat k$ . If $\vec u$ is perpendicular to $\vec a$ and $\vec u \cdot \vec b = 24$ ,then ${\left| {\vec u} \right|^2} = $ . . . .

A line with direction ratios $2,1,2$ meets the lines $x=y+2=z$ and $x+2=2 y=2 z$ respectively at the point $P$ and $Q$. if the length of the perpendicular from the point $(1,2,12)$ to the line $\mathrm{PQ}$ is $l$, then $l^2$ is

If the sum of first $n$ terms of an $A.P.$ is $cn(n -1)$ , where $c \neq 0$ , then sum of the squares of these terms is

The sum of first $20$ terms of the sequence $0.7,0.77,0.777, . . . $ is

The circles ${x^2} + {y^2} - 10x + 16 = 0$ and ${x^2} + {y^2} = {r^2}$ intersect each other in two distinct points, if

Each of the $10$ letters $A,H,I,M,O,T,U,V,W$ and $X$ appears same when looked at in a mirror. They are called symmetric letters. Other letters in the alphabet are asymmetric letters. How many three letters computer passwords can be formed (no repetition allowed) with at least one symmetric letter ?

If $u = {a_1}x + {b_1}y + {c_1} = 0,$ $v = {a_2}x + {b_2}y + {c_2} = 0$ and $\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}},$ then the curve $u + kv = 0$is

Let the lines $y+2 x=\sqrt{11}+7 \sqrt{7}$ and $2 y + x =2 \sqrt{11}+6 \sqrt{7}$ be normal to a circle $C:(x-h)^{2}+(y-k)^{2}=r^{2}$. If the line $\sqrt{11} y -3 x =\frac{5 \sqrt{77}}{3}+11$ is tangent to the circle $C$, then the value of $(5 h-8 k)^{2}+5 r^{2}$ is equal to.......